Magnetic resonance imaging (MRI) provides a powerful tool for non-invasive imaging for treatment assessment and for minimally invasive surgery. The contrast sensitivity of the MRI provides a capability of non-invasively revealing structures and functions of internal tissues and organs not known to other imaging techniques such as, for example, CT scan or Ultrasound.
In MRI data collection and image reconstruction are directly based on Fourier theory. Therefore, the Fourier Transform (FT) forms the basis of the MRI. The underlying Fourier technique is based on the assumption that the sensed signals forming a collected Fourier space—the so-called k-space—contain no temporal changes in spatial frequencies. However, in reality, limitations of equipment, subject motion, respiratory and cardiac activity, blood flow, peristalsis and other physiological fluctuations cause temporal changes in spatial frequencies. In fact, the signals sampled in k-space are actually a subset of (k, t)-space, where t refers to time. Thus, the Fourier technique in MRI actually folds time information with spatial frequency information leading to image distortion and artifacts, which substantially reduce image quality.
Mathematically, the Fourier transform analyzes an entire signal and decomposes the signal into sinusoids of different frequencies. The Fourier transform provides information regarding frequency events within the entire signal. However, the Fourier transform does not provide information regarding the instance of occurrence of a particular frequency component, possibly resulting in the loss of crucial information during signal analysis and processing.
To overcome the deficiency of the FT, other techniques such as the Gabor transform (GT) disclosed in: Gabor, D. “Theory of communications”, J. Inst. Elec. Eng., 1946; 93, 429-457, also known as the short time Fourier transform, and the Wavelet transform (WT) disclosed in: Goupillaud P., Grossmann, A., Morlet J. “Cycle-octave and related transforms in seismic signal analysis”, Geoexplor, 1984; 23, 85-102, and in: Grossmann, A., Morlet J. “Decomposition of Hardy functions into square integrable Wavelets of constant shape”, SIAM J. Math. Anal., 1984; 15, 723-736, have been developed. Both of these methods unfold the time information by localizing the signal in time and calculating its “instantaneous frequencies.” However, both the GT and the WT have limitation substantially reducing their usefulness in the analysis of magnetic resonance signals. The GT has a constant resolution over the entire time-frequency domain which limits the detection of relatively small frequency changes. The WT has variant resolutions, but it provides time versus scale information as opposed to time versus frequency information. Although “scale” is loosely related to “frequency”—low scale corresponds to high frequency and high scale to low frequency—for most wavelets there is no explicit relationship between scale factors and the Fourier frequencies. Therefore, the time-scale representation of a signal is difficult if not impossible to interpret.
It would be advantageous to combine the time-frequency representation of the GT with the multi-scaling feature of the WT in order to overcome the above drawbacks and to provide both time and frequency information by adapting the FT to analyze a localized magnetic resonance signal using frequency dependent time-scaling windows.